Every braid is a half-twist word times a pure braid

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TarchaBraids.braid_corrects_to_pure_range_v1

by WillR · Sep 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

braid-groupsconfiguration-spaceexact-sequencehalf-twistspure-braidstarcha

Every geometric braid can be corrected by the inverse of a word in the explicit elementary half-twists so that the residual braid lies in the image of the ordered-configuration fundamental group, i.e. is pure. This combines adjacent-permutation generation, the explicit deck permutation of each half-twist, permutation correction to the deck kernel, and identification of that kernel with the pure-braid image.

Preamble

import Mathlib
import Definitions.Def_BraidsLinksMCG_ConfigSpace
import Definitions.Def_TarchaBraids_HalfTwist
import Theorems.Thm_TarchaBraids_configProj_isQuotientCoveringMap_v1

Formal statement

namespace TarchaBraids

open BraidsLinksMCG

theorem braid_corrects_to_pure_range_v1 (n : ℕ) (β : GeomBraidGroup n) :
    ∃ w : FreeGroup (Fin (n - 1)),
      β * (FreeGroup.lift (fun i : Fin (n - 1) => halfTwistBraid n i) w)⁻¹ ∈
        (FundamentalGroup.mapOfEq
          ⟨configProj n, (configProj_isQuotientCoveringMap_v1 n).continuous⟩
          (show configProj n (baseOrdered n) = baseUnordered n from rfl)).range := by sorry

end TarchaBraids

Source

Tarcha Teorema 3.11 exact-sequence reduction: match the strand permutation by an adjacent half-twist word, leaving a pure braid.

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